In many biological signal transduction pathways, multiple input signals converge on a shared set of signaling components, which route each input to the appropriate output. How is signaling specificity maintained so that am signal does not corrupt the response of another? For example in yeast, the signals for mating, invasive growth, and osmotic stress are all funneled through the same MAPK (Mitogen Activating Protein Kinase) cascade although each elicits a different response. Here we propose to investigate the dynamics and regulation of signaling cascades through an integrated program of mathematical and experimental approaches. We will develop state-of-the-art mathematical theory and computational tools to analyze and simulate signal transduction pathways, with an emphasis on scaffolding, spatial dynamics, specificity, and how they relate to one another. Our ultimate goal is to develop a theoretical framework for understanding how proper signal processing occurs in highly interconnected biochemical networks and to validate them by detailed modeling and experimentation focusing on the yeast MAPK system. As steps toward this goal, we will first develop generic representations of signaling pathways with shared components and test them in the yeast MAPK system. In this setting, we will rigorously address how scaffolds and feedback regulation can give rise to specificity and what are the limits and tradeoffs. Then, we will include spatial dynamics and explore the implementation of specificity-promoting mechanisms. A hierarchy of models from microscopic levels involving spatial interplay between the scaffold and the resident kinases to a full-scale network level for the yeast MAPK system will be explored. We plan to test our conclusions and predictions from such mathematical and computational analysis by performing selected experiments. The quantitative analysis will involve control theory and large systems of nonlinear ordinary and partial differential equations on networks. New mathematical theories and numerical algorithms will have to be developed for the analysis and simulations.